Experimentation at the Frontiers of Reality in Schubert Calculus
@article{Hillar2009ExperimentationAT, title={Experimentation at the Frontiers of Reality in Schubert Calculus}, author={Christopher J. Hillar and Luis David Garc{\'i}a-Puente and Abraham Mart{\'i}n del Campo and James Ruffo and Zach Teitler and Stephen L. Johnson and Frank Sottile}, journal={arXiv: Algebraic Geometry}, year={2009}, volume={517} }
We describe the setup, design, and execution of a computational experiment utilizing a supercomputer that is helping to formulate and test conjectures in the real Schubert calculus. Largely using machines in instruc- tional computer labs during off-hours and University breaks, it consumed in excess of 350 GigaHertz-years of computing in its first six months of operation, solving over 1.1 billion polynomial systems. This experiment can serve as a model for other large scale mathematical…
11 Citations
Experimentation in the Schubert Calculus
- Mathematics
- 2013
A rich story concerning intrinsic structure, or Galois groups, of Schubert problems is also beginning to emerge from experimentation, which showcases new possibilities for the use of computers in mathematical research.
Lower Bounds in Real Schubert Calculus
- Mathematics
- 2013
We describe a large-scale computational experiment to study structure in the numbers of real solutions to osculating instances of Schubert problems. This investigation uncovered Schubert problems…
REALITY AND COMPUTATION IN SCHUBERT CALCULUS A Dissertation by NICKOLAS JASON HEIN
- Mathematics
- 2013
The Mukhin-Tarasov-Varchenko Theorem (previously the Shapiro Conjecture) asserts that a Schubert problem has all solutions distinct and real if the Schubert varieties involved osculate a rational…
Frontiers of reality in Schubert calculus
- Mathematics
- 2009
The theorem of Mukhin, Tarasov, and Varchenko (formerly the Shapiro conjecture for Grassmannians) asserts that all (a priori complex) solutions to certain geometric problems in the Schubert calculus…
OF REALITY IN SCHUBERT CALCULUS 3 Michael Shapiro in 1993
- Mathematics
- 2009
The theorem of Mukhin, Tarasov, and Varchenko (formerly the Shapiro conjecture for Grassmannians) asserts that all (a priori complex) solutions to certain geometric problems in the Schubert calculus…
The Monotone Secant Conjecture in the Real Schubert Calculus
- MathematicsExp. Math.
- 2015
The monotone secant conjecture posits a rich class of polynomial systems, all of whose solutions are real, and is a compelling generalization of the Shapiro conjecture for Grassmannians.
Solving schubert problems with Littlewood-Richardson homotopies
- MathematicsISSAC
- 2010
A new numerical homotopy continuation algorithm for finding all solutions to Schubert problems on Grassmannians based on Vakil's geometric proof of the Littlewood-Richardson rule is presented.
Combinatorial and Real Algebraic Geometry: Project Description
- Education
- 2018
This proposal is to support my scientific work and that of my advisees in the areas of combinatorial and real algebraic geometry, including related work in applications of algebraic geometry. This…
Disconjugacy and the Secant Conjecture
- Mathematics
- 2015
We discuss the so-called secant conjecture in real algebraic geometry, and show that it follows from another interesting conjecture, about disconjugacy of vector spaces of real polynomials in one…
The Secant Conjecture in the Real Schubert Calculus
- MathematicsExp. Math.
- 2012
We formulate the secant conjecture, which is a generalization of the Shapiro conjecture for Grassmannians. It asserts that an intersection of Schubert varieties in a Grassmannian is transverse with…
References
SHOWING 1-10 OF 68 REFERENCES
Frontiers of reality in Schubert calculus
- Mathematics
- 2009
The theorem of Mukhin, Tarasov, and Varchenko (formerly the Shapiro conjecture for Grassmannians) asserts that all (a priori complex) solutions to certain geometric problems in the Schubert calculus…
Real Schubert Calculus: Polynomial Systems and a Conjecture of Shapiro and Shapiro
- MathematicsExp. Math.
- 2000
This work gives compelling computational evidence for its validity, proves it for infinitely many families of enumerative problems, and shows how a simple version implies more general versions, and presents a counterexample to a general version of their conjecture.
Experimentation and Conjectures in the Real Schubert Calculus for Flag Manifolds
- MathematicsExp. Math.
- 2006
The Shapiro conjecture in the real Schubert calculus, while likely true for Grassmannians, fails to hold for flag manifolds, but in a very interesting way. We give a refinement of the Shapiro…
On reality property of Wronski maps
- Mathematics
- 2007
We prove that if all roots of the discrete Wronskian with step 1 of a set of quasi-exponentials with real bases are real, simple and differ by at least 1, then the complex span of this set of…
Computations in Algebraic Geometry with Macaulay 2
- Computer Science, Mathematics
- 2001
The expositions of the algorithmic tools presented here are designed to serve as a useful guide for those wishing to bring such tools to bear on their own problems and those who are not interested in explicit machine computations at all.
Numerical Evidence for a Conjecture in Real Algebraic Geometry
- MathematicsExp. Math.
- 2000
The optimality of the solving procedure is a promising first step towards the development of numerically stable algorithms for the pole placement problem in linear systems theory.
Enumerative geometry for real varieties
- Mathematics
- 1996
We discuss the problem of whether a given problem in enumerative geometry can have all of its solutions be real. In particular, we describe an approach to problems of this type, and show how this can…
Maximally Inflected Real Rational Curves
- Mathematics
- 2002
We introduce and begin the topological study of real rational plane curves, all of whose inflection points are real. The existence of such curves is a corollary of results in the real Schubert…
Reality and transversality for Schubert calculus in OG (n, 2n+1)
- Mathematics
- 2009
We prove an analogue of the Mukhin-Tarasov-Varchenko theorem (formerly the Shapiro-Shapiro conjecture) for the maximal type B_n orthogonal Grassmannian OG(n,2n+1).